The essentially Σ1 formulae of ΣL are exactly those which are provably equivalent to a disjunction of conjunctions of 2 and Σ1 formulae.

We prove eight necessary and sufficient conditions for a convex quadrilateral to have congruent diagonals, and one dual connection between equidiagonal and orthodiagonal quadrilaterals. Quadrilaterals with both congruent and perpendicular diagonals are also discussed, including a proposal for what they may be called and how to calculate their area in several ways. Finally we derive a cubic equa...

We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes {0, π/4, π/2, 3π/4}. We also prove that four slopes have this property if and only if we can draw K4 with them.

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by L. Schläfli in 1858. Using this, explicit examples of surfaces of every possible type are given.

We outline the small amplitude asymptotic approximation for breathers for one-dimensional chains, and two-dimensional lattices with square, triangular/hexagonal, and honeycomb geometries. Two-dimensional lattices are complicated due to the resulting NLS-type equation being either elliptic or hyperbolic in nature. This gives rise to an additional constraint in addition to the usual condition on ...

We prove the topological expansion for the cubic log-gas partition function